Chapter 5.5, Problem 73PPS

a.

To determine

To write: a function for the area of the basement in terms of x as per the given figure.

Answer to Problem 73PPS

  $A\left(x\right)=8x^2+34x+24$ .

Explanation of Solution

Given:

Alica’s parents want their basement home theater according to the diagram shown at the right.

  

Calculation:

The objective is to express the area in terms of x.

Now, it is to compute the area by adding the areas of the three-rectangle having the dimensions:

  $\begin{array}{l}x+6x+x+\left(x+2\right)+\left(x+2\right)\\ x\cdot x+\left(x+2\right)+\left(x+2\right)\\ x\cdot x+2\end{array}$

Therefore,

  $\begin{array}{l}A\left(x\right)=\left(x+6\right)\left(x+x+x+2+x+2\right)+x\left(x+x+2+x+2\right)+x\left(x+2\right)\\ A\left(x\right)=\left(x+6\right)\left(4x+4\right)+x\left(3x+4\right)+x\left(x+2\right)\{\text{adding like terms}}\\ A\left(x\right)=4x^2+4x+24x+24+3x^2+4x+x^2+2x=0\{opentheparenthesis\text{}}\\ A\left(x\right)=8x^2+34x+24\end{array}$

Hence, a function for the area of the basement in terms of x is $A\left(x\right)=8x^2+34x+24$ .

b.

To determine

To find: the value of x if the basement is to be 1366 square ft.

Answer to Problem 73PPS

  $x=11ft$ .

Explanation of Solution

Given:

Alica’s parents want their basement home theater according to the diagram shown at the right.

Calculation:

From part (a), the equation is $A\left(x\right)=8x^2+34x+24$ .

  $\begin{array}{l}A\left(x\right)=8x^2+34x+24\left\{\text{given polynomial}\right\}\\ 8x^2+34x+24=1366\{\text{given }A\left(x\right)=1366\}\\ 8x^2+34x-1342=0\left\{\text{subtract}1366frombothsides\right\}\\ 4x^2+17x-671=0\left\{\text{divide}2frombothsides\right\}\end{array}$

Now, use the quadratic formula to find the factor:

  $\begin{array}{l}X=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ X=\frac{-17\pm \sqrt{{\left(17\right)}^2-4\left(8\right)\left(-671\right)}}{2\times 4}\left\{\text{substitute these value}\right\}\\ X=\frac{-17\pm 105}{8}\left\{\text{simplify}\right\}\end{array}$

Either $x=\frac{-17+105}{8}$ or $x=\frac{-17-105}{8}$

When $x=\frac{-17+105}{8}$ , then $x=\frac{88}{8}=11$

When $x=\frac{-17-105}{8}$ , then $x=\frac{-122}{8}=-15.25\left\{\text{negative value neglet}\right\}$

Hence, the value of x if the basement is to be 1366 square ft is $x=11ft$ .